#144651

order	512
nilpotency class	4
isoclinism	isoclinic class of #144650
automorphism	?
homology	Abelianization (H₁) = C₂³ · Schur multiplier (H₂) = C₂³

Aliases

No aliases for this group.

Subgroups

	subgroup	G/H
center, socle	C₂	#10392
commutator, Frattini, cosocle	C₂³×Q₈	C₂³
prime radical	C₂⁴	C₂²⋊Q₈
LCS	G ▷ C₂³×Q₈ ▷ C₂³ ▷ C₂ ▷ C₁
UCS	C₁ ◁ C₂ ◁ C₂³ ◁ C₂³×Q₈ ◁ G
LECS	G ▷ C₂³×Q₈ ▷ C₂³ ▷ C₂ ▷ C₁

413 conjugacy classes of subgroups; only the 46 normal subgroups are shown.

normal subgroups only

This group has 413 conjugacy classes of subgroups — too many to draw. Only its 46 normal subgroups are shown.

Dotted edges: containments with no intermediate normal subgroup — omitted non-normal subgroups may lie between.

Full subgroup data →

Presentation

minimal degree	μ(G) = 32
transitive degree	μ_t(G) = 32

A minimal faithful permutation representation — 3 generators on 32 points:

(1,2,5,19,7,12,4,10)(3,13,15,8,17,29,14,25)(6,11,18,27,22,26,20,9)(16,28,30,32,21,23,31,24);(1,3,5,15,7,17,4,14)(2,8,10,13,12,25,19,29)(6,21,18,31,22,16,20,30)(9,23,26,32,27,28,11,24);(1,4,7,5)(2,9,12,27)(3,14,17,15)(6,20,22,18)(8,23,25,28)(10,11,19,26)(13,24,29,32)(16,31,21,30)

Polycyclic presentation:

2^9|a^2=dg,b^2=d,c^2=i,d^2=i,e^2=i,b^a=bd,c^a=ce,d^a=di,e^a=egi,f^a=fghi,g^a=gi,h^a=hi,c^b=cf,e^b=eh,f^b=fg,h^b=hi,d^c=dg,e^c=ei,e^d=ei

Conjugacy classes & character table

32 conjugacy classes — view the full character table.

GAP

Construct this group in GAP via the SmallGroups library:

gap> SmallGroup(512, 51847);